# Fence Patrolling with Two-speed Robots

### Jurek Czyzowicz, Konstantinos Georgiou, Evangelos Kranakis, Fraser MacQuarrie, Dominik Pajak

#### Abstract

A fence is to be patrolled collectively by n robots. At any moment a robot may move in one of the two possible states: walking or patrolling. Each state is associated with a maximal moving speed which cannot be exceeded. We want to schedule the perpetual movements of the robots so as to minimize the idleness, defined as the smallest time interval within which every point is always visited by some robot. First, we give a centralized algorithm constructing schedules with optimal idleness, and subsequently we show an interesting application to a transportation problem concerning Scheduling with Regular Delivery. Our main contribution is the study of distributed, dynamical schedules for patrolling robots with only primitive capabilities. Surprisingly, we are able to design a dynamic schedule for very weak collections of two robots (silent, oblivious, passively mobile), achieving the optimal idleness. Part of our contribution is a very technical analysis of the dynamics of special families of dynamical systems of n robots that we call regular. For such systems we also propose a highly non-trivial O(n2) algorithm to decide whether or not robots converge to a stable configuration thus verifying if the dynamic schedule is optimal.

#### References

- Agmon, N., Kraus, S., and Kaminka, G. A. (2008). Multirobot perimeter patrol in adversarial settings. In ICRA, pages 2339-2345.
- Almeida, A., Ramalho, G., Santana, H., Tedesco, P. A., Menezes, T., Corruble, V., and Chevaleyre, Y. (2004). Recent advances on multi-agent patrolling. In SBIA, pages 474-483.
- Alpern, S., Morton, A., and Papadaki, K. (2009). Optimizing randomized patrols. Operational Research Group, London School of Economics and Political Science.
- Alpern, S., Morton, A., and Papadaki, K. (2011). Patrolling games. Operations research, 59(5):1246-1257.
- Amigoni, F., Basilico, N., Gatti, N., Saporiti, A., and Troiani, S. (2010). Moving game theoretical patrolling strategies from theory to practice: An usarsim simulation. In ICRA, pages 426-431.
- Angluin, D., Aspnes, J., Diamadi, Z., Fischer, M., and Peralta, R. (2006). Computation in networks of passively mobile finite-state sensors. Distributed Computing, 18(4):235-253.
- Angluin, D., Aspnes, J., Eisenstat, D., and Ruppert, E. (2007). The computational power of population protocols. Distributed Computing, 20(4):279-304.
- Beauquier, J., Burman, J., Clement, J., and Kutten, S. (2010). On utilizing speed in networks of mobile agents. In Proceeding of the 29th ACM SIGACTSIGOPS Symposium on Principles of distributed computing, pages 305-314. ACM.
- Chevaleyre, Y. (2004). Theoretical analysis of the multiagent patrolling problem. In IAT, pages 302-308.
- Cieliebak, M., Flocchini, P., Prencipe, G., and Santoro, N. (2012). Distributed computing by mobile robots: Gathering. SIAM J. Comput., 41(4):829-879.
- Collins, A., Czyzowicz, J., Gasieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Martin, R., and Morales Ponce, O. (2013). Optimal patrolling of fragmented boundaries. In SPAA.
- Czyzowicz, J., Gasieniec, L., Kosowski, A., and Kranakis, E. (2011). Boundary patrolling by mobile agents with distinct maximal speeds. Algorithms-ESA 2011, pages 701-712.
- Czyzowicz, J., Kranakis, E., and Pacheco, E. (2013). Localization for a system of colliding robots. In ICALP (2), pages 508-519.
- Dijkstra, E. W. (1982). Selected writings on computing: a personal perspective. Springer-Verlag New York, Inc.
- Dumitrescu, A., Ghosh, A., and Csaba, D. T. (2014). On fence patrolling by mobile agents. CoRR, abs/1401.6070.
- Elmaliach, Y., Agmon, N., and Kaminka, G. A. (2009). Multi-robot area patrol under frequency constraints. Ann. Math. Artif. Intell., 57(3-4):293-320.
- Elmaliach, Y., Shiloni, A., and Kaminka, G. A. (2008). A realistic model of frequency-based multi-robot polyline patrolling. In AAMAS (1), pages 63-70.
- Elor, Y. and Bruckstein, A. M. (2010). Autonomous multiagent cycle based patrolling. In ANTS, pages 119- 130.
- Hare, J., Gupta, S., and Wilson, J. (2015). Decentralized smart sensor scheduling for multiple target tracking for border surveillance. In ICRA, pages 3265-3270. IEEE.
- Hazon, N. and Kaminka, G. A. (2008). On redundancy, efficiency, and robustness in coverage for multiple robots. Robotics and Autonomous Systems, 56(12):1102-1114.
- Kawamura, A. and Kobayashi, Y. (2012). Fence patrolling by mobile agents with distinct speeds. In ISAAC, pages 598-608.
- Machado, A., Ramalho, G., Zucker, J.-D., and Drogoul, A. (2002). Multi-agent patrolling: An empirical analysis of alternative architectures. In MABS, pages 155-170.
- Marden, M. (1949). The Geometry of the Zeros of a Polynomial in a Complex Variable, volume 3 of Math. Surv. AMS.
- Marino, A., Parker, L. E., Antonelli, G., and Caccavale, F. (2009). Behavioral control for multi-robot perimeter patrol: A finite state automata approach. In ICRA, pages 831-836.
- Pasqualetti, F., Franchi, A., and Bullo, F. (2010). On optimal cooperative patrolling. In CDC, pages 7153- 7158.
- Yanovski, V., Wagner, I. A., and Bruckstein, A. M. (2003). A distributed ant algorithm for efficiently patrolling a network. Algorithmica, 37(3):165-186.

#### Paper Citation

#### in Harvard Style

Czyzowicz J., Georgiou K., Kranakis E., MacQuarrie F. and Pajak D. (2016). **Fence Patrolling with Two-speed Robots** . In *Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-171-7, pages 229-241. DOI: 10.5220/0005687102290241

#### in Bibtex Style

@conference{icores16,

author={Jurek Czyzowicz and Konstantinos Georgiou and Evangelos Kranakis and Fraser MacQuarrie and Dominik Pajak},

title={Fence Patrolling with Two-speed Robots},

booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2016},

pages={229-241},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005687102290241},

isbn={978-989-758-171-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - Fence Patrolling with Two-speed Robots

SN - 978-989-758-171-7

AU - Czyzowicz J.

AU - Georgiou K.

AU - Kranakis E.

AU - MacQuarrie F.

AU - Pajak D.

PY - 2016

SP - 229

EP - 241

DO - 10.5220/0005687102290241